OFFSET
1,2
COMMENTS
Conjecture: For any integer m > 0, there are infinitely many positive integers n such that all the gaps prime(n+i+1) - prime(n+i) (i = 0, ..., m-1) are powers of two.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..200
EXAMPLE
a(1) = 1 since all those prime(2) - prime(1) = 3 - 2 = 1, prime(3) - prime(2) = 5 - 3 = 2, prime(4) - prime(3) = 7 - 5 = 2, prime(5) - prime(4) = 11 - 7 = 4, prime(6) - prime(5) = 13 - 11 = 2, prime(7) - prime(6) = 17 - 13 = 4 and prime(8) - prime(7) = 19 - 17 = 2 are powers of two.
MATHEMATICA
PowQ[n_]:=n==2^(IntegerExponent[n, 2])
m=0; Do[Do[If[PowQ[Prime[n+i+1]-Prime[n+i]]==False, Goto[aa]], {i, 0, 7}]; m=m+1; Print[m, " ", n]; Label[aa]; Continue, {n, 1, 10252155}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 24 2014
STATUS
approved